Analysis of Two Parareal Algorithms for Time-Periodic Problems

Gander, Martin J.; Jiang, Yao‐Lin; Song, Bo; Zhang, Hui

doi:10.1137/130909172

Cited by 47 publications

(55 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [13], the so-called PP-PC (periodic parareal algorithm with periodic coarse problem) and PP-IC (periodic parareal algorithm with initial value coarse problem) algorithms are proposed for (3.6). Here, we are interested in PP-PC, because it converges faster than PP-IC.…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Convergence Analysis for Three Parareal Solvers

Wu¹,

Zhou²

2015

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

We analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem u + Au = g. The coarse propagator G is fixed to the backwardEuler method and three time integrators are chosen for the F -propagator: the trapezoidal rule, the third-order diagonal implicit Runge-Kutta (RK) (DIRK) method, and the fourth-order Gauss RK method. It is well known that the Parareal-Euler algorithm using the backward-Euler method for F and G converges rapidly, but less is known when one uses for F the trapezoidal rule, or the fourthorder Gauss RK method, especially when the mesh ratio J(= ΔT /Δt) is small. We show that for a specified λmax(the maximal eigenvalue of A or its upper bound), there exists some critical J cri such that the parareal solvers derived from these three choices of F converge as fast as Parareal-Euler, provided J ≥ J cri . Precisely, for F the trapezoidal rule and the fourth-order Gauss RK method, J cri depends on ΔT, Δt, and λmax and we present concise formulas to calculate J cri , while for F the third-order DIRK method, J cri = 4, independently of these parameters. Numerical examples with applications in fractional PDEs and uncertainty quantification are presented to support the theoretical predictions.1. Introduction. The parareal algorithm, proposed by Lions, Maday, and Turinici in [18], received considerable attention in recent years, because of its potential for parallel-in-time computation. It permits the computation of the solution later in time, before having fully accurate approximations at earlier times, while the global accuracy of the iterative process after a few iterations is comparable to that given by a sequential numerical method used on a fine discretization in time. Nowadays, this algorithm, as well as some other relevant algorithms [5,7,9,10,11,22,29,21], have been used in many fields by many researchers, such as molecular-dynamics simulations [2], morphological transformation simulations [15], structural (fluid) dynamics simulations [5, 10], optimal control [6, 20, 24, 23], Hamiltonian simulations [8,14], simulations of turbulent plasmas [26,27], fast computation of wave equations [7] and Volterra integral equations [17], etc. For a survey of parallel-in-time algorithms, see [16].The algorithm is defined by two time propagators, namely, F and G, which are associated with the small step size Δt and the big step size ΔT . The two step sizes satisfy ΔT = JΔt, where J ≥ 2 is an integer. Since the backward-Euler method is stable for all possible choices of ΔT and the coarse propagator is often forced to take large step sizes, choosing for G the backward-Euler method makes sense. (We can also consider other implicit methods, but they are all more expensive than the backward-Euler method.) Throughout this paper, the G-propagator is fixed to the

show abstract

Section: Resultsmentioning

confidence: 99%

“…Remark 3.3 (relevance to F = e −AΔT ). In the analysis of the parareal algorithm, one often uses the exact propagator F = e −AΔT and this significantly simplifies the convergence analysis; see for example [1,4,12,13,19]. If we use the backward-Euler method as the G-propagator, this simplification yields…”

Section: Resultsmentioning

confidence: 99%

Convergence Analysis for Three Parareal Solvers

Wu¹,

Zhou²

2015

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…In (3.19), the fine propagator is applied to nonperiodic problems and the time periodicity on the coarse time grids will be recovered upon convergence. Then, if G = F =Backward-Euler, following the analysis in [12], the errors e k n and {e k+1 n } also satisfy (2.4)-(2.5). Hence, all the statements for the initial-value problems are also applicable to the PP-PC algorithm (3.19).…”

Section: By Choosingmentioning

confidence: 98%

“…This mean that no speedup is possible with this algorithm, no matter how many processors are used, when the angle θ is 1.2 or larger. [12] proposed the so-called PP-PC (periodic parareal with periodic coarse problem) and PP-IC (periodic parareal with initial value coarse problem) algorithms. Here, we are interested in PP-PC, because in general it converges faster than PP-IC.…”

Section: By Choosingmentioning

confidence: 99%

See 1 more Smart Citation

Convergence Analysis of the Parareal-Euler Algorithm for Systems of ODEs with Complex Eigenvalues

2015

J Sci Comput

View full text Add to dashboard Cite

Parareal is an iterative algorithm and is characterized by two propagators G and F , which are respectively associated with large step size ΔT and small step size Δt, where ΔT = J Δt and J ≥ 2 is an integer. The choice G = F =Backward-Euler denotes the simplest implicit parareal solver, which we call Parareal-Euler, and has been studied widely in recent years. For linear problem U (t) + AU(t) = g(t) with A being a symmetric positive definite matrix, this algorithm converges very fast and the convergence rate is insensitive to the change of J and Δt. However, for the case that the spectrum of A contains complex values, no provable results are available in the literature so far. Previous studies based on numerical plotting show that we can not expect convergence for the Parareal-Euler algorithm on the whole right-hand side of the complex plane. Here, we consider a representative situation: σ (A) ⊆ D(θ ) := {(x, iy) ∈ C : x ≥ 0, |y| ≤ tan(θ )x} with θ ∈ (0, π 2 ), i.e., the spectrum σ (A) is contained in a sectorial region. Spectrum distribution of this type arises naturally for semi-discretizing a wide rang of time-dependent PDEs, e.g., the Fokker-Planck equations. We derive condition, which is independent of J and depends on θ only, to ensure convergence of the Parareal-Euler algorithm. Numerical results for initial value and time-periodic problems are provided to support our theoretical conclusions.

show abstract

Three rapidly convergent parareal solvers with application to time‐dependent PDEs with fractional Laplacian

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

Time‐dependent PDEs with fractional Laplacian ( − Δ)α play a fundamental role in many fields and approximating ( − Δ)α usually leads to ODEs' system like u′(t) + Au(t) = g(t) with A = Qα, where Q∈double-struckRm×m is a sparse symmetric positive definite matrix and α > 0 denotes the fractional order. The parareal algorithm is an ideal solver for this kind of problems, which is iterative and is characterized by two propagators scriptG and scriptF. The propagators scriptG and scriptF are respectively associated with large step size ΔT and small step size Δt, where ΔT = JΔt and J⩾2 is an integer. If we fix the scriptG‐propagator to the Implicit‐Euler method and choose for scriptF some proper Runge–Kutta (RK) methods, such as the second‐order and third‐order singly diagonally implicit RK methods, previous studies show that the convergence factors of the corresponding parareal solvers can satisfy ρ≈13,∀J⩾2 and ∀σ(A)⊂[0,+∞), where σ(A) is the spectrum of the matrix A. In this paper, we show that by choosing these two RK methods as the scriptF‐propagator, the convergence factors can reach 112, provided the one‐stage complex Rosenbrock method is used as the scriptG‐propagator. If we choose for both scriptG and scriptF, the complex Rosenbrock method, we show that the convergence factor of the resulting parareal solver can also reach 112. Numerical results are given to support our theoretical conclusions. Copyright © 2017 John Wiley & Sons, Ltd.

show abstract

Analysis of Two Parareal Algorithms for Time-Periodic Problems

Cited by 47 publications

References 28 publications

Convergence Analysis for Three Parareal Solvers

Convergence Analysis for Three Parareal Solvers

Convergence Analysis of the Parareal-Euler Algorithm for Systems of ODEs with Complex Eigenvalues

Three rapidly convergent parareal solvers with application to time‐dependent PDEs with fractional Laplacian

Contact Info

Product

Resources

About